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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 19, NO. 1, MARCH 1984
The Impact of the Degrees of Operating and
Financial Leverage on Systematic Risk of
Common Stock
Gershon N. Mandelker and S. Ghon Rhee*
I. Introduction
The capital asset pricing model postulates that the equilibrium retum on any risky security is equal to the sum of the risk-free rate of retum and a risk premium measured by the product of the market price of risk and the security's systematic risk. In the capital asset pricing model, beta as an index of systematic risk is the only security-specific parameter that affects the equilibrium retum on a risky se- curity. The identification of the real determinants of the systematic risk of common stock has received a great deal of attention in the finance and accounting litera- ture in recent years. A number of empirical studies have investigated the associa- tion between market-determined and accounting-determined risk measures (see [1], [2], [3], [12], [19], and [23]). These studies have increased our knowledge about correlations between betas of common stock and various accounting vari- ables or accounting betas. The studies cited also have provided further insight into what forms of specification appear to best reduce the measurement errors in estimating accounting betas. In a review of their findings, Foster [10] concludes that the choice of accounting variables has not been guided by a theoretical model linking the firm's financing, investment, and production decisions with its common stock beta. There have been limited efforts to utilize an empirical test design that is more consistent with the definition of beta in the framework of the capital asset pricing model. Under the presumption that the firm's asset stmcture and capital stmcture impact upon operating risk and financial risk, respectively, the separate effect of either financial leverage or operating leverage on beta of common stock has been examined. Hamada [13] reports that approximately one quarter of sys-
*** University of Pittsburgh and University of Rhode Island, respectively. While retaining full responsibility for this paper, the authors would like to thank William Beranek, Dan Givoly, Jeffrey Jaffe, William Margrabe, and anonymous** JFQA referees for their comments on earlier drafts of the paper. The authors also thank J. Rock Chung for his assistance in computer work. Financial support for this paper was provided by the Faculty Research Grant, University of Pittsburgh.
tematic risk is explained by financial leverage while Lev [16] provides empirical evidence that operating leverage, as measured by variable cost, is one of the real determinants of systematic risk. Two recent studies by Hill and Stone [14] and Chance [6] represent more refined applications of the risk decomposition of Ha- mada [13] and Rubinstein [20]. Hill and Stone develop an accounting analogue to Hamada and Rubinstein's formula to investigate the joint impact of operating risk and financial structure on systematic risk. Chance conducts a direct test of the Hamada and Rubinstein formula by controlling operating risk to preserve the assumption of homogeneous risk class. Their findings provide considerable em- pirical support for Hamada and Rubinstein's formula. Recent research efforts further explore the risk decomposition of Hamada and Rubinstein by introducing the degrees of operating and financial leverage into a model that explains betas of common stock. Although the degrees of the two types of leverage are extensively discussed in standard finance textbooks in relation to their impact on the volatility of stockholders' returns or of earnings per share, their relationship with the systematic risk of common stock has not been fully resolved. A recent work by Brenner and Schmidt [5] further extends Rubinstein's analysis of the relationship between the characteristics of the firm's real assets and its common stock beta. They demonstrate how unit sales, fixed costs, contribution margin, and the covariance of sales with returns on the market portfolio affect systematic risk. Gahlon and Gentry [11] show that the beta of a common stock is a function of the degrees of operating and financial leverages, the coefficient of variation of the total revenue, and the coefficient of correlation between earnings after interest and taxes and returns on the market portfolio. Unfortunately, it is difficult to investigate the impact of two types of leverage on operating risk and financial risk in the framework of Gahlon and Gentry. This is so because the degrees of two types of leverage are introduced by an expansion of the coefficient of variation of earnings after interest and taxes. Nonetheless, theoretical analyses of Brenner and Schmidt and Gahlon and Gentry show much promise of enhancing our knowledge of the real determinants of beta. As the degrees of two types of leverage are recognized in a model that identifies the real determinants of beta, this study explores two important empiri- cal issues. First, we examine the joint impact of the degrees of operating and financial leverage on the systematic risk of common stock. Although Hamada and Rubinstein demonstrate that operating risk and financial risk constitute sys- tematic risk, it is not obvious how operating leverage and financial leverage are related to operating risk and financial risk, respectively, in their risk decomposi- tion. We demonstrate how the two types of leverage contribute to systematic risk of common stock. Second, we address the issue of "trade-offs" between operat- ing leverage and financial leverage, while investigating their combined effects on the systematic risk of common stock. The interrelationship between operating and financial leverage is widely discussed in the literature as a means of stabiliz- ing the relative riskiness of stockholders' investment. For example. Van Home ([24], p. 784) states that:
Operating and financial leverage can be combined in a number of different ways to
where 3^ denotes beta of risky corporate debt.^ After a slight rearrangement, we write equation (2) as
(3) P = p + (1 - T)(p -** ^j)D/E.
Financial risk as measured by (1 - T ) ( P * - ^j)DIE causes additional econometric problems associated with a multiplicative effect of financial struc- ture on the beta of risky debt. Although it is not an impossible task to resolve these problems when investigating the real determinants of beta using equation (3), an alternative beta formula is derived to serve our purpose. This formula explicitly incorporates the degrees of operating leverage and financial leverage. By definition, the beta of common stocky is
where Rj, = the rate of retum on common stock j for the period from / - 1 tof, /?„, = the rate of retum on the market portfolio for the period from t
- Hot,
Cov(') and CT ^(•) denote the covariance and variance operators, respec- tively. _ _ Suppose that Rj, = {Ilj,/Ej, _ ]) - 1 where FI^, denotes eamings after inter- est and taxes at t and _Ej, __ , represents the market value of common equity at t
- 1. Substitution of this definition of ^^, into equation (4) yields
We can rearrange equation (5) by multiplying the first argument of the co- variance by Ily, _ 1 /Ily, _ , and subtracting a constant from it
(6) P, = ("-,-./^,-
2 Proposition II of Modigliani and Miller [17], [18] can be expressed as indicated below in the presence of corporate income taxes and risky debt
(a) • E(R) = £(R*) + (1 - T) [ £ ( « * ) - E{R^D/E ,
where ^ = the rate of retum on the levered firm's common stock, R* = therateof retum on the unlevered firm's common stock, and Rji = the rate of retum on risky debt. According to the capital asset pricing model, £(R) = R^ -V [£(«„) - R/]p, E(k^ = Rf + [E{RJ - Rj]^,mdE{Rj) = R^ + [E{RJ - «^]p^, whereR^= the rate of retum on a risk-free asset. Substitution of these expressions into (a) yields
The degree of financial leverage (DFL) is defined as the percentage change in n that results from a percentage change in X, where X denotes earnings before interest and taxes. Thus,
(7) DFL =[(n,/n,_.)-i]/[(W .)-
Solving for (fly,/ 11^, _ i) - 1, we have
The degree of operating leverage (DOL) is measured by the percentage change in X that is associated with a given percentage change in the units pro- duced and sold.3 Let Q denote the number of units. Thus,
Solving for _(.Xj,/Xj,i) - 1, we obtain
Successive substitution of (10) into (8) and (8) into (6) yields
(11) p. =
Let S denote sales in dollars. Thus, S = pQ where p is the price per unit. By multiplying the first argument of the covariance in ( l i ) by p/p, we obtain the desired result
(12) P^. = (DOL) (DFL) p° ,
where pP = Cov[(n^,,/S^,,)(V^;'-i)' ^m<]/^~ (^m,)' Note that Ily,. i/Sj,_ , represents the net profit margin at r - 1 while Sj,/Ej,_ , measures the
3 When the units produced and the units sold differ due to an uncertain demand for the products, stochastic cost-volume-profit analysis is introduced. (See [15] and [21].) '' It is important to note that both DOL and DFL are not random variables. For example. DOL as defined by (9) can be modified as D O L =
where p = the price per unit. V = the variable cost per unit, and F = the total fixed costs. Equation (a) represents another definition of DOL that indicates its nonrandomness. Likewise. DFL as defined by (7) can be rewritten as
^ ' = [(p- '^Qj.- 1 - /=;,- • ] / [(p - '^Qj- 1 - pj,-. - 'j,- 1].
where / denotes interest expenses.
units produced and sold rather than annual sales in dollars. Because the quantity produced and sold is not available from the income statement, following Lev [16], we use annual sales as a proxy as indicated by (13)." Estimation procedures based on (13) and (14) rest on the restrictive, ceteris paribus, assumption of stationary elasticity over the estimation period. To examine the assumption of stationarity, the Chow [7] and Fisher [9] test is conducted for each firm based on regressions over two subperiods, t, = 1957-1966 and t,, = 1967-1976. The test results indicate that we cannot reject the hypothesis that the degrees of two types of leverage are stable for approximately 90 percent of the firms at a = 5 percent. One possible option available would be to choose only those firms that pass the Chow-Fisher test but this would reduce the size of the sample. Since we employ a portfolio-grouping approach that should lessen the degree of nonstationarity of the coefficients, we decided not to eliminate any of the firms in our sample.^-^ The following market model is used to estimate the beta of each common stock. The measurement of monthly rates of retum on the market portfolio is based on a value-weighted index of the New York Stock Exchange stocks com- piled by CRSP. ~ ;• = 1- (15) Rj, = a. + P / , , + V., , ^ ^
where v^, denotes a disturbance term. Table 1 summarizes estimates of beta, the degree of operating leverage, and the degree of financial leverage for 255 firms in the sample by industry. The 255 firms are distributed over 10 different industries under the 2-digit SIC Industry Code.
B. Regression Results We investigate the combined effects of the degrees of two types of leverage on systematic risk by using the following equation
(16) Lnpp = 7o + 7i LnDOL^ + 7
where P_, DOL^, and DFL_ are portfolio means of beta, the degree of operating leverage, and the degree of^financial leverage. A portfolio-grouping approach is
- When negative earnings are observed for either X or 11. the following regressions are run without a logarithmic transformation Xj, = <(., -f .t.^^,, + 8.,. and n., = <,^ + ^jA^, + i.,.
After (^2 and tji^ are estiinated. a;in (13) is approximated by 2{Sj/Xj) and dj^in (14)_is approximated by <lj2(Xy/7Ij) where Sj. Xj. and iI denote the 20-year average values of S,,, Xj,. and Jlj,. •' Lev [16] uses the annual sales as a proxy for the units produced and^sold.
- In his examination of the assumption of stationarity. Lev [16] compares the estimates from regressions for tbe whole period and a subperiod and concludes tbat tbe differences in estimated coefficients are minimal. He does not use any statistical test to support his findings. ' When those firms that did not pass the Chow-Fisher test were excluded from our sample, we found tbat tbe overall results did not improve much from wbat is reported in tbe empirical portion of this paper.
TABLE 1 Estimates of Average Beta, DOL, and DFL by Industry
Industry Code
2000 2600 2800 2900 3200 3300 3500 3600 3700 4900
(Food and Kindred) (Paper & Allied Products) (Chennical) (Petro-Chemical) (Glass & Cement Gypsum) (Steel) (Machinery) (Appliances) (Auto) (Utilities)
Number of Firms 32 13 36 20 10 21 40 24 24 35
Beta
DOL*
DFL*
- DOL = the Degree of Operating Leverage.
- DFL = the Degree of Financial Leverage.
employed to reduce the errors-in-variables bias.'" Under this grouping proce- dure, we rank the sample firms on the basis of the size of DOL in ascending order. We place the first five securities in portfolio 1, the next five in portfolio 2, and the last five securities in portfolio 51. The average p, DOL, and DFL are calculated for each portfolio, respectively. The same procedures are used to form 51 portfolios based upon the size of DFL. We also group the sample firms on the basis of the size of P to investigate whether or not firms with higher betas show greater trade-offs between DOL and DFL than firms with lower betas. One has to recognize a potential selection bias because the grouping and cross-sectional regressions are performed in the same study period. To correct this bias, we introduce instmmental variables which should be highly correlated with the two independent variables but which can be observed independently of the two." The natural candidates for our purpose would be operating leverage and financial leverage measured in book values. Out of several proxies available, we choose the 20-year average of the ratio of net fixed assets to total assets and the ratio of total debt to total assets as appropriate instmmental variables for DOL and DFL, respectively.
Table 2 presents test results of the hypothesis that both DOL and DFL have positive effects on the beta of common stock. The table presents the cross-sec- tional regression estimates based on three sets of data. Each data set has 51 port- folios that are formed from rankings of two instmmental variables for DOL and DFL, and beta, respectively. For each set of data, three regression results are reported. The coefficients in the first lines in each panel show the association between the portfolio's beta and both DOL and DFL. The second and third lines report the results when either DOL or DFL is suppressed.
The empirical results are consistent with the hypothesized relationship: re- gression coefficients of DOL and DFL are consistently positive, suggesting that both are positively associated with the relative riskiness of common stock. The explanatory power of both operating and financial leverage is quite high, ranging
'" See [4], [8], and [2] for details about such grouping procedures and their statistical merits. " See [22], p. 445.
regression results in the last panel show that DFL demonstrates much higher ex- planatory power than does DOL, 32 percent versus 0.3 percent, when the magni- tude of DFL is used for rankings of common stocks to form portfolios. Consider- ing the ranking method employed, it is not surprising. For example, when ranking is done according to DFL, we have 51 portfolios, each with various levels of DOL. Therefore, when DOL is used as an independent variable in the regression, we would indeed expect it to have a small explanatory power. A sim- ilar phenomenon would be observed when ranking is done on the basis of DOL while DFL is used as an independent variable in the regression. As reported in the top two panels of Table 2, however, the same phenomena do not occur when instrumental variables are used for rankings of common stocks. When regression coefficients are estimated using 51 portfolios formed based upon rankings of beta, we find that DFL alone can explain as much as 33 percent of cross-sectional variation of betas and DOL alone explains 14 percent,'^ Because of limited data, we have not included an independent variable representing the intrinsic business risk of common stock. The estimates of the intercept that are significant in all regressions appear to capture the infiuence of this omitted variable. Furthermore, the intercept's estimates seem to be stable from one regression to another.
C. Tests of the Trade-Off Hypothesis between Operating Leverage and Financial Leverage
The second hypothesis to be examined is the relationship between DOL and DFL. It has been proposed in the literature that management tries to stabilize the level of the beta of common stock. Frequent changes in the beta of common stock, so it is argued, impose transaction costs on stockholders because they have to rebalance their portfolios to maintain them at a desired level of risk. The de- gree of operating leverage is an important factor to be considered in the firm's asset structure decisions. By changing from a labor-intensive manufacturing pro- cess to a capital-intensive one, a significant change would occur in the cost struc- ture of the firm. A rise in fixed costs and a simultaneous decline in variable cost per unit increase the degree of operating leverage and thereby increase the rela- tive riskiness of common stocks. However, the firm's decision on the operating leverage can be offset by its decision on its financial leverage. To save portfolio revision costs to the stockholders, the two types of leverage can be chosen so that changes in the level of beta are minimized. If the level of intrinsic business risk is constant, a change in DOL can be offset by a change in DFL and vice versa. Therefore, one would expect a cross-sectional negative correlation between DOL and DFL.
'2 When the cross-sectional regression is performed at the level of the individual firm, the fol- lowing results are obtained
Pj = ,05 + ,14Ln DOL,. + ,44Ln DFL_. R^ = , (3-3O)| (3,13) |, (4.92) where figures in parentheses are t-values. They are statistically significant at a = 1 percent. The smaller R ^ reported for the regression can be attributed to measurement errors of variables at the level of the individual firm,
Tahle 3 presents the estimated correlation coefficients for the 51 portfolios formed from rankings of operating leverage, financial leverage, and heta, respec- tively. As expected, we observe consistent negative correlations hetween DOL and DFL. Negative correlations are particularly pronounced when either operat- ing leverage or financial leverage is used for ranking. The respective correlations are pCDOL^, DFL^) = - .30 and - .32 for the whole sample. These correla- tions are significant at a = 1 percent. When portfolios are formed on the basis of the rankings of beta, we observe a negative and nonsignificant correlation be- tween the two types of leverage, p(DOLp,DFLp) = - .05, for the whole sam- ple. To investigate why this happens, we divide the portfolios into two subgroups, one group with low betas and another with high betas. It appears that firms with high betas engage in trade-offs more actively than do firms with low betas.
Operating Leverage
Financial Leverage
Beta
Low
High
Whole
Low
High
Whole
Low
High
Whole
TABLE 3 Test Results of the Trade-off Hypothesis
Number of Portfolios 25
26
51
25
26
51
25
26
51
Beta
[10]
[.11]
[.11]
[.12]
[10]
[.12] . [.14]
[.17] 106 [.25]
DOL . [.17]
[21] . [28] . [.10] . [13] . [.11] . [•10]
[14] . [13]
DFL . [.07] . [06] . [.06] . [13]
[.08] . [.14] . [.06]
[.06] . [.07]
p(DOLp,DFLp) -. (1.26) -. (1.18) -. (2.19)t -. (1.61)# -. (1.59)# -. (2.39)t -. ( .37) -. (2.73)t -. ( -35) Figures in parentheses are t-values. t Statistically significant at a = 1 percent.
Statistically significant at a = 10 percent.
Figures in brackets are cross-sectional standard deviations.
The average DOL and DFL of the two subgroups also provide some evi- dence of balancing activities between the degrees of two types of leverage. For portfolios formed based upon rankings of DOL, it appears that low DOL is com- bined with high DFL, .73 versus .99, and vice versa, 1.16 versus .97. The same
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